1
FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007
Lecture 3 – Univariate and Bivariate
Probability
Agenda
É
Simulation of Exponential Random Variable
É
Simulation of Uniform Random Variable
É
Simulation of Gumbel Random Variable
É
Non-Parametric Density Models
É
Bivariate Continuous Distributions
É
Multivariate Continuous Distributions
É
Bivariate Discrete Distributions
É
Multivariate Discrete Distributions
2
FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007
Simulation of Exponential Random Variable
We use the inversion of CDF transformation from
Lecture 2 to generate sample random variables or
deviates that are independent and identically
distributed (i.i.d.) according the following
exponential distribution
(
)
(
)
(
)
1
exp
X
F
x
x
H
x
λ
⎡
⎤
=
−
−
⎣
⎦
We define the following relationship
(
)
1
X
X
F
U
−
=
3
FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007
Simulation of Exponential Random Variable
(
)
(
)
(
)
(
)
1
1
exp
exp
1
log 1
log 1
U
X
X
U
X
U
X
U
λ
λ
λ
λ
−
=
−
−
−
=
−
−
=
−
= −
−
Excel workbook file Class03a.xls contains the
sample simulation.
R workspace contains the
corresponding simulation using R.
Let’s estimate
the mean and variance of an exponential random
variable.

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